Commuting fraction equals five-eighths iff inner automorphism group is Klein four-group

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Statement

The following are equivalent for a finite group G:

  1. The Commuting fraction (?) of G is exactly equal to 5/8.
  2. The Inner automorphism group (?) of G has order equal to four.
  3. The Inner automorphism group (?) of G is a Klein four-group.

The condition defines a family of isoclinic groups. The groups in this family that are 2-groups form the \Gamma_2 family in the Hall-Senior nomenclature.

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